Local circular law for random matrices

Paul Bourgade Harvard University Horng-Tzer Yau Harvard University Jun Yin University of Wisconsin-Madison

Probability mathscidoc:1608.28011

Probability Theory & Related Fields, 159, (3), 545-595, 2013
The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z|−1|≥τ for arbitrarily small τ>0, the circular law is valid around z up to scale $N^{-1/2+ \e}$ for any $\e > 0$ under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.
Local circular law Universality
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@inproceedings{paul2013local,
  title={Local circular law for random matrices},
  author={Paul Bourgade, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823143443254928406},
  booktitle={Probability Theory & Related Fields},
  volume={159},
  number={3},
  pages={545-595},
  year={2013},
}
Paul Bourgade, Horng-Tzer Yau, and Jun Yin. Local circular law for random matrices. 2013. Vol. 159. In Probability Theory & Related Fields. pp.545-595. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823143443254928406.
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